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A084950 Array of coefficients of denominator polynomials of the n-th approximation of the continued fraction x/(1+x/(2+x/(3+..., related to Laguerre polynomial coefficients. 15
1, 1, 2, 1, 6, 4, 24, 18, 1, 120, 96, 9, 720, 600, 72, 1, 5040, 4320, 600, 16, 40320, 35280, 5400, 200, 1, 362880, 322560, 52920, 2400, 25, 3628800, 3265920, 564480, 29400, 450, 1, 39916800, 36288000, 6531840, 376320, 7350, 36, 479001600, 439084800, 81648000, 5080320, 117600, 882, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

A factorial triangle, with row sums A001040(n+1), n >= 0.

Conjecture: also coefficient triangle of the denominators of the (n-th convergents to) the Continued Fraction w/(1+w/(2+w/3+w/... This C.F. converges to 0.697774657964... = BesselI[1,2]/BesselI[0,2] for w=1. - Wouter Meeussen, Aug 08 2010

For general w, Bill Gosper showed it equals n! 2F3((1/2-n/2,-n/2); (1,-n,-n); 4*w). - Wouter Meeussen, Jan 05 2013

From Wolfdieter Lang, Mar 02 2013: (Start)

The row length sequence of this array is 1 + floor(n/2) = A008619(n), n >= 0.

The continued fraction 0 + K_{k=1}^{infinity}(x/k) = x/(1+x/(2+x/(3+... has n-th approximation  P(n,x)/Q(n,x). These polynomials satisfy the recurrence q(n,x) = n*q(n-1,x) + x*q(n-2,x), for q replaced by P or Q with inputs P(-1,x) = 1, P(0,x) = 0 and Q(-1,x) = 0 and Q(0,1) = 1. The present array provides the Q-coefficients: Q(n,x) = sum(a(n,m)*x^m, m=0 .. floor(n/2)), n >= 0. For the P(n,x)/x coefficients see the companion array A221913. This proves the first part of W. Meeussen's conjecture given above.

The solution with input q(-1,x) = a and q(0,x) = b is then, due to linearity, q(a,b;n,x) = a*P(n,x) + b*Q(n,x).  The motivation to look at the q(n,x) recurrence came from an e-mails from Gary Detlefs, who considered integer x and various inputs and gave explicit formulas.

This array coincides with the SW-NE diagonals of the unsigned Laguerre polynomial coefficient triangle |A021009|.

The entries a(n,m) have a combinatorial interpretation in terms of certain so-called labeled Morse code polynomials using dots (length 1) and dashes (of length 2). a(n,m) is the number of possibilities to decorate the n positions 1,2,...,n with m dashes, m from {0, 1, ..., floor(n/2)}, and n-2*m dots. A dot at position k has a weight k and each dash between two neighboring positions has a label x. a(n,m) is the sum of these labeled Morse codes with m dashes after the label x^m has been divided out. E.g., a(5,2) = 5 + 3 + 1 = 9 from the 3 codes: dash dash dot, dash dot dash, and dot dash dash, or (12)(34)5, (12)3(45) and 1(23)(45) with labels (which are in general multiplicative) 5*x^2, 3*x^2 and 1*x^2 , respectively. For the array of these labeled Morse code coefficients see A221915. See the Graham et al. reference, p. 302, on Euler's continuants and Morse code.

Row sums Q(n,1) = A001040(n+1), n >= 0. Alternating row sums Q(n,-1) =  A058797(n).  (End)

For fixed x the limit of the continued fraction K_{k=1}^{infinity}(x/k) (see above) can be computed from the large order n behavior of Phat(n,x) and Q(n,x) given in the formula section in terms of Bessel functions. This follows with the well-known large n behavior of BesselI and BesselK, as given, e.g., in the Sidi and Hoggan reference, eqs. (1.1) and (1.2). See also the book by Olver, ch. 10, 7, p. 374. This continued fraction converges for fixed x to sqrt(x)*BesselI(1,2*sqrt(x))/BesselI(0,2*sqrt(x)). - Wolfdieter Lang, Mar 07 2013

REFERENCES

Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics, 2nd ed.; Addison-Wesley, 1994.

F. W. J. Olver, Asymptotics and Special Functions, Academic Press, 1974 (1991 5th printing).

LINKS

Table of n, a(n) for n=0..48.

Avram Sidi and Philip E. Hogan, Asymptotics of modified Bessel functions of high order. Int. J. of Pure and Appl. Maths. 71 No. 3 (2011) 481-498

FORMULA

Given a flush left array, (r, c) of row (n+1) = (n+1)*(r-1, c) + (r-2, c-1); (i.e. multiply each term of n-th row by (n+1) and add the first term on the NW diagonal; getting the terms in row n+1.

(n-m)!/m!*binomial(n-m,m). - Wouter Meeussen, Aug 08 2010

From Wolfdieter Lang, Mar 02 2013: (Start)

Recurrence (short version): a(n,m) = n*a(n-1,m) + a(n-2,m-1), n>=1, a(0,0) =1, a(n,-1) = 0, a(n,m) = 0 if n < 2*m. From the recurrence for the Q(n,x) polynomials given in a comment above.

Recurrence (long version): a(n,m) = (2*(n-m)-1)*a(n-1,m) + a(n-2,m-1) - (n-m-1)^2*a(n-2,m), n >= 1, a(0,0) =1, a(n,-1) = 0,  a(n,m) =0 if n < 2*m. From the standard three term recurrence for the unsigned orthogonal Laguerre polynomials. This recurrence can be simplified to the preceding one, because of the explicit factorial formula given above which follows from the one for the Laguerre coefficients (which, in turn, derives from the Rodrigues formula and the Leibniz rule). This proves the relation a(n,m) = |Lhat(n-m,m)|, with the coefficients |Lhat(n,m)| = |A021009(n,m)| of the unsigned n!*L(n,x) Laguerre polynomials.

For the e.g.f.s of the column sequences see  A021009 (here with different offset, which could be obtained by integration).

E.g.f. for row polynomials gQ(z,x) := sum(Q(n,x)*z^n, z=0 .. infty) = (I*Pi*sqrt(x)/sqrt(1-z))*(BesselJ(1, 2*I*sqrt(x)*sqrt(1-z))*BesselY(0, 2*I*sqrt(x)) - BesselY(1, 2*I*sqrt(x)*sqrt(1-z))*BesselJ(0,2*I*sqrt(x))), with the imaginary unit I = sqrt(-1) and Bessel functions. (End)

The row polynomials are Q(n,x) = Pi*(z/2)^(n+1)*(BesselY(0,z)*BesselJ(n+1,z) - BesselJ(0,z)*BesselY(n+1,z)) with z := -I*2*sqrt(x), and the imaginary unit I. An alternative form is Q(n,x) = 2*(w/2)^(n+1)*(BesselI(0,w)*BesselK(n+1,w) - BesselK(0,w)*BesselI(n+1,w)*(-1)^(n+1)) with w := -2*sqrt(x). See A221913 for the derivation based on Abramowitz-Stegun's Handbook. - Wolfdieter Lang, Mar 06 2013

limit(Q(n,x)/n!,n -> infinity) = BesselI(0,2*sqrt(x)). See a comment on asymptotics above. - Wolfdieter Lang, Mar 07 2013

EXAMPLE

The irregular triangle a(n,m) begins:

n\m          0          1        2        3      4    5  6 ...

O:           1

1:           1

2:           2          1

3:           6          4

4:          24         18        1

5:         120         96        9

6:         720        600       72        1

7:        5040       4320      600       16

8:       40320      35280     5400      200      1

9:      362880     322560    52920     2400     25

10:    3628800    3265920   564480    29400    450    1

11:   39916800   36288000  6531840   376320   7350   36

12:  479001600  439084800 81648000  5080320 117600  882  1

...Reformatted and extended by Wolfdieter Lang, Mar 02 2013.

E.g. to get row 7, multiply each term of row 6 by 7, then add the term NW of term in row 6: 5040 = (7)(720); 4320 = (7)(600) + 20; 600 = (7)(72) + 96; 16 = (7)(1) + 9. Thus row 7 = 5040 4320 600 16 with a sum of 9976 = a(7) of A001040.

The denominator of w/(1 + w/(2 + w/(3 + w/(4 + w/5)))) equals 120 + 96w + 9w^2. - Wouter Meeussen, Aug 08 2010

From Wolfdieter Lang, Mar 02 2013: (Start)

Recurrence (short version): a(7,2) = 7*72 + 96 = 600.

Recurrence (long version): a(7,2) = (2*5-1)*72 + 96 - (5-1)^2*9 = 600.

a(7,2) = binomial(5,2)*5!/2! = 10*3*4*5 = 600. (End)

MATHEMATICA

Table[CoefficientList[Denominator[Together[Fold[w/(#2+#1) &, Infinity, Reverse @ Table[k, {k, 1, n}]]]], w], {n, 16}]; or equivalently Table[( (n-m)!*Binomial[n-m, m] )/m! , {n, 0, 15}, {m, 0, Floor[n/2]}] (* Wouter Meeussen, Aug 08 2010 *)

CROSSREFS

Cf. A001040, A180047, A180048, A180049.

Cf.  A021009 (Laguerre triangle). For the A-numbers of the column sequences see the Cf. section of A021009. A221913.

Cf. A052119.

Sequence in context: A005299 A185586 A128728 * A180317 A066654 A145960

Adjacent sequences:  A084947 A084948 A084949 * A084951 A084952 A084953

KEYWORD

tabf,nonn,easy

AUTHOR

Gary W. Adamson, Jun 14 2003

EXTENSIONS

Rows 12 to 17 added based on formula, Wouter Meeussen, Aug 08 2010

Name changed by Wolfdieter Lang, Mar 02 2013

STATUS

approved

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Last modified July 22 18:10 EDT 2017. Contains 289671 sequences.